Using computer simulations we study how a corona of polymer molecules grafted to cellulose nanocrystal aggregate (CNA) particles influences the interaction between pairs of parallel CNAs. The resulting distance and orientation (face-to-face versus edge-to-edge) dependence is very rich and counterintuitive. Although the unperturbed polymer corona assumes cylindrical symmetry relatively quickly as the degree of polymerisation increases, the polymer mediated interactions between the grafted particles are strongly orientation dependent. Rather unexpectedly we find that the forces in the face-to-face orientation are much larger than in the edge-to-edge configuration although in the latter case the distance between the particle surfaces is much smaller. The reason for this effect is that overall the face-to-face orientation leads to larger chain confinement. Interestingly, we find that the deviations of the polymer mediated interactions from cylindrical symmetry are larger in the case of longer grafted molecules compared to shorter ones. When the distance between the CNAs becomes larger and the overlap of the polymer coronas becomes small, the orientation dependence of the mediated interaction vanishes and the particles behave as cylindrical rods. However, this is only a crossover point where the behaviour of the system inverts to slightly larger forces in the edge-to-edge compared to the face-to-face configuration. Thus, even though the polymer density around the CNAs is nearly perfectly cylindrically symmetric the polymer mediated interactions are strongly orientation dependent, revealing the polygon character of the CNA cross-section.